Birational Weyl group actions and q-Painleve equations via mutation combinatorics in cluster algebras

A cluster algebra is an algebraic structure generated by operations of a quiver (a directed graph) called the mutations and their associated simple birational mappings. By using a graph-combinatorial approach, we present a systematic way to derive a tropical, i.e. subtraction-free birational, representation of Weyl groups from cluster algebras. Our results provide an extensive class of Weyl group actions, including previously known examples with algebro-geometric background, and hence are relevant to the q-Painleve equations and their higher-order extensions. Key ingredients of the argument are the combinatorial aspects of the reflection associated with a cycle subgraph in the quiver. We also study symplectic structures of the discrete dynamical systems thus obtained. The normal form of a skew-symmetric integer matrix allows us to choose Darboux coordinates while preserving the birationality.